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arXiv:quant-ph/0305003 · Physical Review A

Bound entangled states violate a nonsymmetric local uncertainty relation

Holger Friedrich Hofmann

Published Sep 22, 2003• 40 citations
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Abstract

As a consequence of having a positive partial transpose, bound entangled states lack many of the properties otherwise associated with entanglement. It is therefore interesting to identify properties that distinguish bound entangled states from separable states. In this paper, it is shown that some bound entangled states violate a nonsymmetric class of local uncertainty relations [H. F. Hofmann and S. Takeuchi, Phys. Rev. A 68, 032103 (2003)]. This result indicates that the asymmetry of nonclassical correlations may be a characteristic feature of bound entanglement.

DOI
10.1103/physreva.68.034307

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As a consequence of having a positive partial transpose, bound entangled states lack many of the properties otherwise associated with entanglement. It is therefore interesting to identify properties that distinguish bound entangled states from separable states. In this paper, it is shown that some bound entangled states violate a nonsymmetric class of local uncertainty relations [H. F. Hofmann and S. Takeuchi, Phys. Rev. A 68, 032103 (2003)]. This result indicates that the asymmetry of nonclassical correlations may be a characteristic feature of bound entanglement.
The efficient experimental verification of entanglement requires an identification of the essential physical properties that distinguish entangled states from non-entangled states. Since the most characteristic feature of entanglement is the extreme precision of correlations between spatially separated systems, we propose a quantitative criterion based on local uncertainty relations (quant-ph/0212090). Some basic sum uncertainty relations for N-level systems are introduced and the amount of entanglement that can be verified by violations of the corresponding local uncertainty limit is discussed.
We propose an entanglement criterion based on local uncertainty relations (LURs) in a stronger form than the original LUR criterion introduced by Hofmann and Takeuchi [H. F. Hofmann and S. Takeuchi, Phys. Rev. A 68, 032103 (2003)]. Using arbitrarily chosen operators ${{\mathrm{A\ifmmode \hat{}\else \^{}\fi{}}}_{k}}$ and ${{\mathrm{B\ifmmode \hat{}\else \^{}\fi{}}}_{k}}$ of subsystems A and B, the tighter LUR criterion, which may be used not only for discrete variables but also for continuous variables, can detect more entangled states than the original criterion.
We provide operational definition of asymmetry of entanglement: An entangled state contains asymmetric entanglement if its subsystems can not be exchanged (swapped) by means of local operations and classical communication. We show that in general states have asymmetric entanglement. This allows to construct nonsymmetric measure of entanglement, and a parameter that reports asymmetry of entanglement contents of quantum state. We propose asymptotic measure of asymmetry of entanglement, and show that states for which it is nonzero, contain necessarily bound entanglement.
We explore the link between two concepts: the level of violation of a Bell inequality by a quantum state and discrimination between two states by means of restricted classes of operations, such as local operations and classical communication and separable ones. For any bipartite Bell inequality, we show that its value on a given quantum state cannot exceed the classical bound by more than the maximal quantum violation shrunk by a factor related to the distinguishability of this state from the separable set by means of some restricted class of operations. We then consider the general scenarios where the parties are allowed to perform a local preprocessing of many copies of the state before the Bell test (asymptotic and hidden-nonlocality scenarios). We define the asymptotic relative entropy of nonlocality and, for positive partial transpose states, we bound this quantity by the relative entropy of entanglement of the partially transposed state. The bounds are strong enough to limit the use of certain states containing a private key in the device-independent scenario.
The positivity of the partial transpose is in general only a necessary condition for separability. There exist quantum states that are not separable, but nevertheless are positive under the partial transpose. States of this type are known as bound entangled states, meaning that these states are entangled but they do not allow distillation of pure entanglement by means of local operations and classical communication. We present a parametrization of a class of $2\ifmmode\times\else\texttimes\fi{}2$ bound entangled Gaussian states for bipartite continuous-variable quantum systems with two modes on each side. We propose an experimental protocol for preparing a particular bound entangled state in quantum optics. We then discuss the robustness properties of this protocol with respect to the occupation number of thermal inputs and the degrees of squeezing.